Dynamics of Atmospheres and Oceans, 13 (1989) 269- 300 269 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
O C E A N M O D E L L I N G S T U D I E S IN S U P P O R T O F A L T I M E T R Y
PAULA BERRY and JOHN MARSHALL
Space and Atmospheric Physics Group, Department of Physics, and Centre for Remote Sensing, Imperial College, London SW7 2BZ (Gt. Britain)
(Received February 18, 1988; revised August 22, 1988; accepted September 9, 1988)
ABSTRACT
Berry, P. and Marshall, J., 1989. Ocean modelling studies in support of altimetry. Dyn. Atmos. Oceans, 13:269-300
Simulation studies are presented that consider the degree to which baroclinic ocean circulation models are constrained by observations of the surface pressure field on space and time-scales possible with future satellite-borne altimeters. Two- and three-layer eddy resolv- ing quasi-geostrophic models are used to simulate and assimilate altimetric observations.
For synoptic mapping of the oceanic eddy field on the scale of an ocean basin, a satellite orbit repeat time of 14 days is found to be optimum. Our studies suggest that altimetric observations will provide strong constraints on the surface flow of such models and, if used in conjunction with them, may be used to infer the deep flows.
1. INTRODUCTION
Satelli te a l t imet ry offers the p rospec t of global coverage of surface geo- s t rophic cur ren ts on synopt ic t ime-scales, and as such could p lay a cent ra l role in fu ture oceanograph ic observing campaigns . Perhaps the only way of ma k ing sense of the i n f o r m a t i o n p rov ided by a l t imeters and b lend ing it in a dynamica l ly cons is ten t way wi th o ther types of data , is b y ass imila t ion in to dynamica l ocean mode ls in m u c h the same way as meteoro logis t s f o rm their analyses.
Ass imi la t ion of global da t a sets in to global ocean models is no t a realistic possibi l i ty at present . F o r the immed ia t e fu ture we mus t p roceed in stages invest igat ing regional p rob lems with ocean models ta i lored to the prevai l ing local dynamics . In the presen t con t r i bu t i on some aspects of synopt ic map- p ing of geos t rophic eddies f r om a l t imet ry are cons ide red b y ass imila t ion in to a mul t i - layer quas i -geos t rophic ocean model .
A synoptic description of the oceanic geostrophic eddy field has as its aim a sequence of instantaneous maps at high spatial (10-100 km) and temporal (weeks-months) resolution. The altimeter has the potential to provide an appropriate data set for mapping the surface signature of the oceanic eddy field. This particular application of altimetry is considered here because we believe it to be a feasible one, rather than because it should necessarily be regarded as having a high scientific priority. The geostrophic eddy field has a strong surface-height signal and is of small horizontal scale and so is
/
associated with pronounced sea-surface tilts. Systematic long wavelength uncertainties in tides and satellite orbits, which may compromise our ability to monitor the gyre-scale circulation, may not be limiting on the synoptic-scale. Further, detailed gravimetric geoids exist for the western North Atlantic whose errors do not appear to swamp the oceanographic signal, suggesting that absolute velocity determination may be a possibility (see Cheney and Marsh, 1981). Finally, since the geostrophic eddy field is a dynamical instability of the larger scale flow, and is rather well described by the conservation of potential vorticity on synoptic time-scales, our dynami- cal understanding is arguably sufficient to contemplate the use of numerical models to assimilate the height-field data.
A less ambitious, but nevertheless valuable application of models lies in observing system simulation studies where they can be used to optimize the deployment of observing platforms. For example, in altimetry there needs to be a compromise between spatial and temporal coverage. Models can help us arrive at opt imum specifications for, say, synoptic mapping of Gulf Stream variability.
Our investigation assesses the degree to which observations of the surface pressure field on the space and time-scales possible with a satellite-borne radar altimeter are useful in constraining eddy resolving, basin-scale ocean circulation models. Our study is the first, we believe, to contemplate assimilation into basin-scale models that simultaneously resolve the synoptic eddy field. Observing system simulation studies are presented which suggest that such data will provide strong constraints on the surface flow in a baroclinic model, and allow us to infer the deep flow.
In section 2 the rationale for using dynamical models as a vehicle for blending diverse data types is set out, stressing the importance of the model as a source of a priori information. The objective analysis technique em- ployed to blend model and data is introduced. In section 3 an ocean model is briefly described and in section 4 observing system simulation studies with two and three-layer versions of the model are presented. A mechanism by which information at the surface is projected vertically is proposed and provides the dynamical background used to interpret the data assimilation experiments.
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2. THE USE OF MODELS IN THE ANALYSIS OF DATA
Oceanographic data are of many types, of varying quality and is irregu- larly distributed in space and time. The observing system is improving but it will always be incomplete and subject to errors, much more so than the meteorological network, for in the ocean there are almost insurmountable sampling problems. From these observations a best estimate of the true state of the ocean is to be constructed. Because of the sparcity of the data and their inaccuracy there are an infinite number of ocean states consistent with the observations and so there is no alternative but to rely heavily on a priori information in deciding which ocean state is most likely. Minimum variance estimation, a tried and tested method of optimally combining observations with a priori information, is now outlined in a context that is familiar to oceanographers and in a manner which makes transparent the use of prior knowledge (for a very useful introduction to the theory see Ghil et. al. (1981)). The technique will be employed to blend model and data in our simulation studies of sections 3 and 4.
2.1. Minimum variance estimation
Suppose u are variables representing the state of the ocean on a regular grid (dimension n) and there are incomplete and regularly spaced observa- tions of u, z (dimension - -p , p < n), contaminated by errors
z = A u + c (1 )
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For example, in Fig. I (showing the upper layer stream function from an eddy-resolving model), the u would correspond to the values of the stream- function at the grid points in each layer of the numerical model and z would correspond to the (simulated) satellite observations of the upper layer along the selected tracks indicated by the criss-crossing black lines. Since some function of the variable may be observed rather than the analysis variable itself, the entries of A need not be 0 or 1. However, in our simple case A is made up of only 0s and l s and has a particularly simple form reflecting the regular spatial pat tern of the (repeated) satellite tracks. From these observa- tions, z, our object is to make the best estimate of u, fl.
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to give an estimate of u, h. To derive an opt imum H (in the sense of least squares) we form the covariance of the estimated error (from eqns. (1) and (2)) and choose the one that minimizes its trace. This opt imum H is (see, for example, Rodgers (1976) or Ghil et al. (1981) and references therein)
H = (ATCZ'A + Cu-')-IATc~ - ' (3)
the well known min imum variance estimator, where C, is the covariance of u, and C c is the covariance of the observing errors.
Of all the linear estimators, eqn. (3) is the best in the sense of least squares. Substituting eqn. (3) into eqn. (2), gives an estimate
fl = C, AT(AC, A T + C~)- lz
= (ATC~-'A + Cul)-IATc(-lz (4)
with an expected error
C a _ , = (ATC~-'A + C=a) -1 (5)
Equation (4) is useful if we are willing to make a priori assertions concerning the spatial correlations and amplitude of the field u (i.e., if there is a priori knowledge of Cu), for it describes how to weight the observations in an opt imum way. How good the best estimate is depends on our ability to make useful statements about the statistics of u and the observational errors (within the limitations of the data).
If, however, we are not prepared to make any statements about the solution covariance at all, but have information only about the statistics of the observational errors (C, is assumed known) then eqn. (4) reduces to (as Cu--, o0)
fl = (ATC(- aA) - 1ATC2 lz (6)
Equation (6) will be useful in section 2.2, when a priori information, for example as provided by an ocean model, is treated as data.
In most situations, though, the minimum of information solution (eqn. 6)) can be improved because there is a priori information about the variance and spatial correlation of z. This comes from past observations and dynami- cal understanding and can be used to make useful statements about C, enabling our estimate to proceed via eqn. (4).
In the present context our object is to synoptically map the geostrophic eddy field exploiting altimetry and an ocean model. On synoptic time-scales the dynamics is governed by the close conservation of potential vorticity (see section 3). The preferred scale of ocean eddies is on the Rossby radius of deformation, L = NHe/f, where H e is the equivalent depth of a vertical
274
mode (first baroclinic, say), N is a measure of the stratification and f is the Coriolis parameter. Even though our knowledge of geostrophic eddy dy- namics may be incomplete and our statements therefore not very precise, it is nevertheless useful information with which to guide our choice of C u. In this way, prior knowledge of ocean dynamics is brought to bear in the analysis of the observations, constraining the number of possible solutions.
For instance, a particularly simple, yet useful, form of the correlation is the Gaussian covariance structure function
F ( ~ ) = e-'2~/b)2
where F(~) = ~ [ u ( r ) - fi][u(r + ~ ) - f i ] ) and fi is the mean. F(~) is the average over all r of two values of u separated by a distance ~. The correlation scale, b, is set by the spatial scale of the field u, of the order of the Rossby radius for the oceanic synoptic scale.
It is important that the covariance structure function be chosen carefully, since it determines the scales in the data that are enhanced in the estimate, and also how the information is interpolated between data points. The way in which the estimator H weights the various scales in the data can be made transparent by looking at the eigenvectors and eigenvalues of Cu; this is discussed further in, for example, Rodgers (1976), Hollingsworth (1984) and, in the present context, by Marshall (1985a). Using such techniques, it can be shown that if a Gaussian covariance is chosen for C~, and this is combined with a diagonal covariance C, = o 21 for random, uncorrelated error, H will always smooth the data, the extent to which it does so being dependent on b.
2.2. Models as a source of a priori information
The minimum variance estimator (eqn. (4)) is very useful if our a priori information is only of a statistical nature, but a potentially more powerful and direct use of our dynamical understanding would be to use a dynamical model to provide a priori information about the field, u, in addition to its statistics, i.e., to use it as a 'first guess'. This will be of particular value in the oceanographic context because the interpolation problems are so acute that all our prior knowledge of ocean dynamics must be brought to bear in the interpolation procedure to make the best use of the data available. Since the ocean model is the most concise statement of this knowledge, it seems natural to use it as an analysis tool. In such an approach the ocean model provides a first guess of u, which we will call fJ( - ), adopting the notation of Ghil et al. (1981). It is enlightening to regard the first guess as a 'vir tual observation' that is of the same nature as the field observations, but which differs only in its error characteristics; for a more detailed discussion from
275
this perspective see Marshall (1985a). Virtual observations can be treated explicitly as data by augment ing the data eqn. (1) thus
f J ( - ) = Iu + ¢~ (7)
Here the ocean model is being used as a measuring ins t rument providing a first guess of u, with an error % that is assumed independent of the observations. When observations z of u become available they are combined with a first guess to form a better estimate of u, fi( + ). So eqns. (1) and (7) are combined to give (following Jackson, 1979)
z , = A , u + ¢ , (8)
where
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The covariance of data errors Ce. = E(ccT,) is
C ' * = ( C~O U0)
where U = E(%¢ T) is the covariance of the ocean model errors and C, = E(¢¢ T) is the covariance of errors in the observations.
Equat ion (8) demonstra tes clearly how the use of a priori informat ion has changed the original underdc te rmined problem, eqn. (1), into an overde- termined problem (n + p > n) with a unique solution. Because our a priori informat ion has been treated as data, eqn. (8) can be solved using eqn. (6) to give a new, improved estimate
fl( + ) = (ATC~- aA + U-a)-I[ATC~-az + U - I l l ( - ) ] (9a)
with expected error
U + = (ATC~-IA + U - a ) -a (9b)
This is the most general form of m i n i m u m variance estimator. Equat ion (5) is a special case of eqn. (9) with fl( - ) = 0.
Before proceeding further it should be emphasized that all a priori informat ion can be treated as virtual observations (dynamical constraints, smoothness criteria, climatology, etc.) by writing it in the form (8). This is an enlightening perspective f rom which to view a priori information. In eqn. (5) the 'v i r tual observation' was f l ( - ) = 0 and the expected error in this observat ion was U.
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Although at first sight eqn. (9) looks rather complicated, it is nothing more than a straightforward generalization of the well-known scalar formula
Y'. (observation/error in observation) estimate =
Y] (1/error)
Not surprisingly the best estimate weights the observations (real and virtual) with respect to their error covariances.
2.3. Optimum interpolation
Equation (9) involves the inversion of large (n x n) matrices and so is of limited practical use. However it can be written in a less computationally taxing form thus (see, for example, Rodgers (1976))
f i (+) = f i ( - ) + K [ z - Aft(-)] (lOa)
where
K = UAT(AUA T + C,) -~ (10b)
is a matrix of weighting coefficients. Unlike eqn. (9), eqn. (10) only involves the inversion of ( p Xp) matrices, where p is the number of observations. It relates the correction to the first guess to observed deviations from it, through the matrix K. Information about the geometrical configuration of the observation and estimation points is contained in the covariances, and our choice of U determines how the data are interpolated spatially. Equation (10) has been derived on numerous occasions and applied in many diverse fields. It is the inverse method used by Wunsch (1978) to determine the ocean circulation from hydrographic sections. Equation (10) is the general form in which linear estimation is applied in meteorological interpolation, where it is known as objective analysis. It was first introduced into the meteorological literature by Eliassen (1954) and later developed by Gandin (1965). The most systematic and concerted effort to apply objective tech- niques combining dynamical models and data in an oceanic context is that due to Robinson and collaborators (see Robinson (1986) for a review).
Equation (10) sets out a general procedure to combine observations and a priori constraints optimally. We now describe in more detail how the formalism could be used to blend observations of ocean topography from altimetry with virtual observations provided by an eddy resolving model to synoptically map the geostrophic eddy field. As in the meteorological application of eqn. (10), instead of adopting climatology as a first guess, a model is used to provide a short-range forecast interpolating information in space and time. In this way the forecast provides information about the
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field, and so can be regarded as a measuring instrument just like the altimeter. The philosophy behind the use of a dynamical model in this way is based on the assumption that the information provided by it is close in some sense to the truth (i.e., the model has some skill). It is then reasonable to linearize the inversion about the first guess as assumed in eqn. (10a).
3. OBSERVING SYSTEM SIMULATION STUDIES
3.1. The ocean model
We briefly describe an eddy-resolving circulation model of the Gulf Stream and its recirculation, which is used to establish a ' t ruth ' or ' refer- ence' circulation. This is sampled to simulate an altimetric data set and a degraded version of the model is subsequently employed to assimilate the data.
The ocean is an N layer quasi-geostrophic model confined to a rectangu- lar basin on a #-plane, and governed by the quasi-geostrophic potential vorticity equation. Data assimilation experiments have been carried out with two- and three-layer versions of the model. The circulation is driven by an imposed wind-stress curl and frictionally retarded
0q - - + J ( + , q) = G (11) 0t
where q is the quasi-geostrophic potential vorticity, ~b is the streamfunction, and G are potential vorticity sources and sinks.
A finite-difference version of eqn. (11) is stepped forward from a known initial state in a double gyre configuration
O < _ x < L
- 0 . 5 L _ < y _ < 0 .5L
with
~7 A(wind stress) = T o sin(2~ry/L)
to generate a reference 6-month sequence of 'synoptic ' maps of the Gulf Stream and its recirculation. Details of the numerical implementat ion can be found in Brugge et al. (1987). Model parameters are given in the Appendix.
Figure 1 shows the streamfunction (a) and potential vorticity (b) in the uppermost layer of a two-layer integration in a 3000 x 3000 km basin. An unstable internal jet flows eastward parallel to the front in the potential vorticity field, separating subpolar and subtropical gyres. The scale of the eddying motion is set by the Rossby radius of deformation ( - 48 km); the eddies have a strong pressure signal and so are associated with pronounced
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sea-surface slopes. In Fig. 1 the sea-surface drops by about 1 m, moving northwards across the eastward flowing Gulf Stream. The time-scale of the eddying motion is of the order of weeks and months; cut-off warm and cold rings with closed potential vorticity contours (indicating material transfer of fluid across the front) are particularly long lived. Inspection of grey-scale movie sequences of the potential vorticity field reveals a wealth of dynamical activity: there is evidence of Rossby wave radiation from the meandering Gulf Stream, Rossby wave breaking, ring formation, spontaneous generation of vortex pairs, etc. In common with all quasi-geostrophic models there is perhaps less of a proclivity for loops to pinch-off from the Stream: as can be seen in Fig. lb the internal jet remains rather contiguous. Furthermore, in this two-layer integration the eddy activity tends to be concentrated in the region of the internal jet and is not prevalent in the interior of the gyre. However, in the experiments presented in section 4.5, with a three-layer model, the interior of the gyre is also explicitly unstable and the eddy activity more uniformly distributed over the basin.
Despite the limitations of the quasi-geostrophic formulation, it has several advantages over other perhaps more realistic models with more degrees of freedom. If it is assumed (consistently with the layered formulation) that the horizontal velocity does not vary with depth within the layer, then the upper layer streamfunction can be interpreted in terms of the ocean topography, the deviation of the sea surface from the gravimetric geoid h = ( f / g ) + , where f is the Coriolis parameter and g is the acceleration due to gravity. Thus the model can assimilate height field data in an obvious way. Further- more, because it is a filtered model, it does not suffer from initialization problems associated with spurious gravity wave activity generated by noisy data. Finally, the model has space and time-scales in common with the oceanic synoptic scale; the flow depicted in Fig. 1 has many realistic features both in the mean and eddy characteristics.
3.2. Sampling strategy
On reaching a statistically steady state, the model is integrated forward to generate a reference 6-month sequence of 'synoptic' maps of the eddying Gulf Stream and its recirculation. This is our ' truth' circulation, which is then sampled to simulate the altimetric data set.
The ' truth' circulation is sampled, simulating the spatial density and temporal frequency of the tracks laid down by a satellite in an exactly repeating orbit. For such an orbit, with period ~'rev and repeat time T, there are T/'rre v revolutions per repeat cycle, giving a track spacing at the Equator of
360 d - - - (degrees)
T/~rev
TABLE I
Repeat times and track spacings for a satellite in an 800-km-altitude orbit
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Repeat time Track separation at 45 o North d / 2 b a
(days) (d (km))
10 200 1.3 14 140 0.9 30 63 0.4
a b is the correlation scale (78 km) used in the simulation experiments.
Thus the temporal and spatial resolutions are limited by
d x T--- 360~'re v
For SEASAT, which flew at an altitude of 800 km, ~'rev = 0.0698 days, yielding the following approximate relationship between repeat-time and track spacing
repeat time x spacing between tracks at equator - 24 (12) (days) (degrees)
The spacing between the tracks decreases with the cosine of the latitude. In our simulation studies the satellite is constrained to pass over columns
and rows of the finite-difference grid, giving an independent measurement at each grid point along the track. To give an impression of the spatial resolution possible, simulated tracks on a 14-day repeat over the 3000 x 3000 km basin are shown in Fig. 1. Table I gives chosen track separations, d, as a function of repeat time. The satellite is flown over this grid, laying down its tracks in a pre-determined order; subsequent experiments were found to be insensitive to the details of this ordering.
It is assumed that after the required corrections (due to sea-state bias, atmospheric water vapour, atmospheric loading, ionospheric effects, etc.) have been made, there remains an irreducible random error, c, in the sea-surface height measurement. It is supposed that errors due to orbital uncertainties and tides have sufficiently long wavelengths relative to the synoptic scale that they result in a systematic error along the track that can be eliminated. It should be emphasized that this is perhaps a rather optimis- tic assumption on the scale of the gyre since long-wavelength tracking errors may be difficult to separate from the mean circulation on scales of 3000-4000 km. Furthermore, no attempt is made to simulate the error in the gravimet- ric geoid, but we acknowledge that it is this spatially correlated error (of uncertain amplitude and scale), rather than the random errors, which will ultimately limit the potential of altimetry in ocean circulation studies. The combined problem of determining the ocean circulation whilst improving
280
the geoid, using ocean models in conjunction with altimetry, is discussed by Marshall (1985a).
Thus, our simulated observations of the deviation of the sea surface from the gravimetric geoid along a track are
Z = g track -[- ~ (13)
where h t rack is taken from the ' t ruth ' circulation and c is the random error component taken from a normal distribution with zero mean and standard deviation o 0
Expectation E(ccT) = o2I } (14)
and E(c) = 0
Although eqn. (14) is a gross oversimplification of the expected errors (principally geoid and tracking) of the altimetric system, and will lead to over-optimistic results, it seems sensible here to adopt the simplest possible error structure to provide a base-line for future reference. For example it is by no means clear, a priori, that perfect observations of the surface pressure field on the space and time scales possible with a satellite-borne altimeter can constrain a highly non-linear eddy-resolving ocean circulation model such as the one employed here.
3.3. Analysis procedure
We now consider the problem of forming an estimate ]1 over the entire I × J grid from simulated observations repeated along selected tracks. As stressed in section 2 this is an underdetermined problem and so heavy reliance must be placed on a priori information.
A climatological field at the resolution of the grid could be chosen as a first guess together with an estimate of its error. But this is an understate- ment of our a priori knowledge because there is dynamical information that can be brought to bear in forming the best estimate. Having simulated the data by sampling an evolving flow governed by eqn. (11), it would be sensible to assimilate the observations into the same model. The model can then be employed to advect information between the repeated satellite tracks, thus projecting past information into the future. One should be wary, though, of only assimilating data into a perfect model. In reality there are many dynamical and physical processes that are not adequately described by an ocean model, and so there is a danger of over-optimistic results if the model that simulated the data is also used to assimilate it. To represent this model imperfection crudely, in section 4.4 degraded versions of eqn. (11) are also used to assimilate the data.
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Thus the observations are augmented with an a priori estimate of h that would have been made in the absence of altimetry
z.-- h( - )
As discussed in section 2, the 'first guess' and the observations are optimally combined applying the least-squares procedures eqn. (10) (weighting with respect to error covariances of z and h ( - )).
The following continuous updating strategy is employed: (1) starting from a prescribed climatological initial state; (2) the model is integrated forward in time providing an estimate of the
ocean topography h ( - ) , which is in error because of inadequacies in the model physics and the imperfectly known initial state;
(3) when the satellite flies overhead the observations z along a particular track are optimally combined with the estimate of the ocean topography currently carried by the model, h ( - ) , to provide a (hopefully) better estimate of the true state of the ocean, h( +) ;
(4) the model is then integrated forward until the next observation time, when it again provides the first guess in step 3.
In this way, the model is used to extrapolate the information provided by the altimeter forward in time, whilst our objective mapping techniques interpolate between the tracks. The relative merits of various updating and observing strategies can be measured with reference to the truth circulation by, for example as chosen here, computing the root-mean-square-error
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(r.m.s.e.) between h and h. Other objective measures can be chosen (see, for example, Rienecker et al. (1987)) but would not lead to qualitatively differ- ent conclusions.
Because the model is continuously updated and the satellite tracks are straight, the analysis problem eqn. (10) reduces to a series of one-dimen- sional ones. Each time the satellite passes overhead, estimates along and either side of the track are made. Fifteen data points equally spaced every 15 km along the track (at the resolution of the numerical model) are used to analyse for the ocean topography. The radius of influence of the data is determined by the correlation scale of the covariance matrix of the model first guess errors, chosen to be a Gaussian covariance structure function (see section 2). At distances from the track much greater than the correlation scale b, the analysis tends to the first guess, h (+ ) - - , h ( - ) . We choose to update the model fields on either side of the track out to a distance of 2b.
The performance of the analysis is rather sensitive to the choice of correlation scale. If b is too large, the analysed fields are too smooth and fail to capture the details of rings and eddies and frontal structures. Alterna- tively, if the correlation scale is too small, the impact of data insertion can
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be detrimental since spurious, small-scale features are introduced into the analysis, which very rapidly approaches the first guess on either side of the satellite track. In our studies the opt imum b was found empirically to be 78 km, somewhat less than twice the Rossby radius. This correlation scale was kept fixed and applied over the whole basin and at all times. It is doubtful that this assumption of stationary error covariance structure function is strictly valid in such a non-linear, non-stationary eddy field. More com- plicated (and realistic) error models could have been constructed, but the added computational effort involved in the implementation of such a scheme would not, in our opinion, have led to substantially improved results.
4. A L T 1 M E T R I C D A T A AS A C O N S T R A I N T ON B A R O C L I N I C C I R C U L A T I O N M O D E L S
4.1. Repeat-time strategies
Figure 2 shows the r.m.s.e, between h and tl as a function of time for various repeat-time strategies in the two-layer integration from which Fig. 1 is taken. For clarity only the upper layer curves are displayed. The impact of data on lower layers is considered in section 4.3. Such r.m.s.e, curves are often normalized with respect to the climatological variability. Here, how- ever, it was thought that more information was conveyed by presenting the errors in absolute units: to normalize they should be compared with the no-data integration (curve A) which gives a measure of the natural variabil- ity. For each repeat-time the same amount of data (in toto) are supplied, only its spatial and temporal frequency differs (see eqn. (12)). Data were assimilated into the model (an identical twin, or perfect model) starting from a 'climatological' initial state, taken to be a linear Stommel (1948) solution in the upper layer of the model, with no flow beneath. Our results are not sensitive to our choice of initial conditions.
When no data are inser ted, curve A, the r.m.s.e, between h and ]l remains fairly steady at 8 cm. This is a useful reference since it is a measure of the variability of the model about climatology. It should be remembered that this is a basin-scale measure: the variance is concentrated in the region of the separated jet where it rises to as high as 30 cm r.m.s..
Assimilation of perfect data into the model on 10-, 14- and 30-day repeat cycles reduces the r.m.s.e, to about one-third of the natural variability about climatology. This in itself is a significant result since it demonstrates that observations from a single (albeit perfect) altimetric system are capable of constraining a two-layer eddy resolving basin-scale ocean circulation model. Although data are assimilated by the model continuously, the repeat fre-
283
10-
~ 4 E L)
UJ CO
D
-6
E 0 0 0
v
r 3
r~
0 20 40 60 80 100 120 (0 ) TIME (days)
. .
(b) Fig. 2. (a) Roo t -mean-square error (r.m.s.e) be tween the t ru th and assimilat ion model ocean topographies f rom a two-layer model initialized with a Stommel solution in the upper layer an d no mot ion benea th : Curve A, no data assimilated; curve B, data assimilated f rom a 10-day repeat orbit ; curve C, data assimilated from a 14-day repeat orbit ; curve D, data assimilated f rom a 30-day repeat orbit . Curve A is a measure of the variabil i ty of the model abou t the l inear solution. (b) Analysis of upper layer s t reamfunct ion (14-day repeat) on day
108; CI = 5 cm of ocean topography.
10-
quencies can be seen in the r.m.s.e, curves. This is because the error falls dramatically as the satellite flies over the model Gulf Stream where most of the variance is concentrated. Data in the quiescent interior region reduce the r.m.s.e, locally, but have only a small impact on the error in toto. Such considerations explain why, with a 30-day repeat (curve D), the error undergoes large fluctuations during its cycle. The satellite spends too much of its time in regions where the variance is small, allowing the model to deviate further from the truth.
A useful way of rationalizing the trade-off between space and time resolution is to consider the ratio d/2b, where d is the track separation, and b is the correlation scale of the signal (see Table 1). Our experiments suggest that an opt imum sampling strategy would make this ratio d/2b - 1. Indeed the r.m.s.e, is reduced to a minimum for a repeat-time of 14 days (curve C in Fig. 2) implying a track separation of 140 km, almost exactly twice the correlation scale b. This result was obtained earlier by Marshall (1985b), but by using a barotropically unstable one-layer ocean circulation model in a miniature ocean basin.
To give a visual impression of the success of the assimilation procedure the analysis of the upper layer streamfunction on day 108 of the assimilation using data from a 14-day repeat orbit is shown in Fig. 2b. A comparison with the truth for the same time (Fig. la) shows that the major features of the circulation are reproduced rather effectively in the analysis, and the upper layer flow is well constrained by data from one altimeter. Evidently,
no noise
~ 4 E o
~ 2 rr
284
-9
6
E o o o x
3 tJ 69
0 20 40 60 80 100 120 TIME (days)
Fig. 3. E r ro r curves for the ocean t o p o g r a p h y ob ta ined by ass imi la t ing da ta wi th vary ing
noise levels on a 14-day repeat into a two-layer model initialized as described in Fig. 2.
285
in a 30-day repeat-cycle, the satellite over-samples in space (d/2b = 0.4) at the expense of temporal resolution; the converse is true in a 10-day repeat orbit (d/2b = 1.6).
4.2. Impact of random data errors
Figure 3 demonstrates the insensitivity of the analysis error to (reasona- ble) random errors in the observations. In each case the repeat time is 14 days. A random error of 8 cm (r.m.s.) (a signal to noise ratio of unity) does not cause a significant deterioration in the analysis, since a Gaussian covariance results in a very effective smoothing weights matrix K. It should be no surprise that the analysis error can be less than the random error in each independent measurement because the interpolation procedure tends to average the observations over a correlation scale. Further, a wealth of a priori information is being brought to bear on the analysis through the use of eqn. (11), which encapsulates our dynamical understanding of the field.
4.3. Extrapolation of single-level data in the vertical
Given that remotely sensed data can only give direct information about properties of the ocean surface, one of the most important objectives of our study is to ascertain to what extent a dynamical model can be used in conjunction with altimetry to infer the subsurface flow. Aspects of this problem have been considered by Hurlbert (1985) in a limited-area model of the Gulf of Mexico.
Figure 4 presents the r.m.s.e, between the truth and analysis ~k2 fields obtained by assimilating perfect data on a 14-day repeat into a two-layer perfect model initialized as described in section 4.1. Note that now, for this
E 0
x LU 03
cr
t0=14days - ~ B
C
0 20 40 6 0 8 0 100 120 TIME (days )
Fig. 4. Lower layer r.m.s.e, curves for the two-layer integration in Fig. 2. Curve A, r.m.s.e, in ~b z when no data are assimilated; curve B, r.m.s.e, in q~2 when perfect data are assimilated on a 14-day repeat; curve C, Aq~2exp[-- c(t -- to)]: t o =14 days, and hq,2(to) is the r.m.s.e, in 4'2 at t o .
Jnoluo~ o~oz oql s! ou!i )I~|q )l~.tql O
q~L "t s zu~ 00~l=
ID '-801_ K
~p uo su
o!~
lu!
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A~I (E
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ep-~[) uo!l~i!tu!ss~ (q) pu~
qln~l (e) tuo~j suo!lounjtu~o~ls ~oK
g I ~OA
~O'I "~ "~t!~ I
(q)
J),1
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287
subsurface layer, the vertical scale is expressed in dimensional streamfunc- tion units of m 2 s - ' . No vertical extrapolation procedures are employed: rather the lower layer flow is allowed to evolve freely, adjusting to the continually updated surface flow. The lower layer r.m.s.e, is markedly reduced over a period of 100 days or so; inspection of the instantaneous flow patterns reveals that the position and scale of the eddy field is reproduced with some skill (see Fig. 5, which presents the lower layer truth and analysed fields after 60 days).
A plausible dynamical mechanism that accounts for the success repre- sented by the curve in Fig. 4 is now emerging.
Energetic eddies in the upper layer (in and above the main thermocline) drive the deep flow through vertical motions, which act like 'plungers' rippling the interface on the scale of 100-200 km, generating Rossby waves in the lower layer (see, for example, Rhines and Holland (1979) or Haidvo- gel and Rhines (1983)). The lower layer of the two-layer model is deep and the currents weak and so non-linear processes do not dominate.
Accordingly, suppose that the lower layer vorticity equation can be linearized thus
3 262 + -~x62 = -¢x7 262 + 3W12 (15) 0t V
where ~1 is the fractional upper layer depth and E is the coefficient of bot tom friction, then wl2, the vertical velocity at the interface acts as a 'wave-maker' associated with the baroclinic instability of the model Gulf Stream and recirculation. The forcing field w12 is a strongly varying func- tion of space and time consequent on the rapidly evolving, highly non-linear eddies in the upper layer. It can be expressed in terms of an 'omega' equation thus
2w12- H*w 2 = + - ( a , - G)
+ -~x (~b, - ~b2) (16)
where R is a Rossby number of the vorticity equation, R = Us/~L 2, (Us is a characteristic horizontal velocity, and L is the basin width, see Appendix), H* = H2/H1H2 and -y = L J L where Lp = (g'H1H2/fo2H) 1/2 is the Rossby radius and g ' is the reduced gravity. However, because the deep flow is weak, the right-hand side of eqn. (16) does not depend strongly on ~b 2. Given a good analysis of the upper level synoptic-scale pressure field (proportional to ~bl) the spatial and temporal variation of the vertical velocity can be deduced from eqn. (16) with imperfect knowledge of the qJ2 field.
288
Equation (15) suggests that, given 14212 from the 'omega' equation, the error in the lower layer streamfunction will propagate as a damped Rossby w a v e
a a - - V 2 A•2 -~- - - A + 2 = - - c V 2 A+2 (17) at ax
where Aq, 2 = ~pv z - -+~, the truth minus the analysis streamfunctions. Equa- tion (17) has solutions
A~ba(t ) ---- A ~ 2 ( t 0 ) e - ' ( t - '°) (18)
Thus any error in the initial conditions will decay with an e-folding time of 1 / c - - 5 5 days, for the bot tom friction coefficient used in our two-layer integration. The dashed line in Fig. 4 is the curve A + z ( t ) = A~bz(t0)e -~(t- '°) where t o = day 14 of the assimilation. By this time, after one repeat cycle, the error in the upper level streamfunction has dropped significantly, and so the w12 forcing the lower layer is quite close to the truth. The falling exponential curve models the decrease in the r.m.s.e, rather well, showing that the error in the lower layer streamfunction is well represented by a superposition of damped linear Rossby waves.
In order to further test our hypothesis, the effects of incomplete observa- tions were eliminated by initializing the lower layer with no flow, as in Fig. 4, but replacing the upper layer streamfunction with the truth at all times over the whole basin ( ' infinite altimetry'). As shown in Fig. 6. the decrease in the lower layer r.m.s.e, is then given almost exactly by eqn. (18), this time with t o = 0.
4.4. Degraded models
Here a degraded model is used to assimilate our altimeter data. Two degradations will be considered, representing uncertainties in -f and R. We
E o o o
6 O
¢ r , x
0 20 40 60 80 100 120 TIME (days)
Fig. 6. Lower layer r.m.s.e, in a two-layer integration, initialized as in Fig. 2, in which the upper layer is replaced by the truth at all times over the whole grid (' infinite altimetry'). The crosses are values of the function A~2(t = 0) e x p ( - ct).
289
~ 4 E
laJ o9 ~ 2 . tr
-6
E o o
3 LU ¢/)
rr
0 20 40 60 80 100 120 TIME (days)
Fig. 7. Er ro r curves for the ocean t o p o g r a p h y (in cm) and ~b 2 (in m 2 s - 1 ) o b t a i n e d w h e n pe r f ec t d a t a f rom an o p t i m u m orb i t repea t cycle (14 days) is ass imi la ted in to a two- layer
m o d e l in w h i c h the s t ra t i f ica t ion is in error ; g' = 2gttruth ( in i t ia l izat ion as in Fig. 2).
will see that their impact on the assimilation process as a constraint on the surface flow, and the ability of the model to interpolate vertically, can be understood in terms of the mechanism of transfer of information into the lower layer presented in the previous section. The degraded two-layer models were initialized with a Stommel solution in the upper layer, and zero motion beneath, and altimeter data from a satellite in an opt imum orbit (repeat time 14 days, track spacing 140 km) assimilated as described previously. The resulting r.m.s.e, curves can therefore be compared directly with those in Fig. 2 (curve C) and Fig. 4, where a perfect model was used.
In the first degraded model, the density jump between the two layers has been doubled, giving g ' = 4 × 10 -2, increasing the Rossby radius Lp and hence 7 by a factor of 212; one might expect the assimilating model to produce anomalously large eddies. If altimetry is to constrain the upper layer circulation, then data must be supplied at a sufficiently high rate to compensate for this tendency.
The r.m.s.e, curves between the truth and degraded assimilation models shown in Fig. 7. for both upper and lower layer flows. They do not depart far from the r.m.s.e, curves shown in Fig. 2, indicating that our ability to deduce the surface flow from altimetry has not been significantly impaired when assimilating into a model whose stratification is in error. Thus it appears that if data are supplied on a 14-day repeat strategy, the rate of supply of information is sufficiently high to offset this particular model degradation.
290
A change in the stratification of the magnitude considered here has only a marginal impact on the ability of the model to interpolate vertically; the lower layer curve in Fig. 7. shows a slight increase in the r.m.s.e, over that obtained with a perfect model (cf. Fig. 4, curve B). This is consistent with our 'omega' eqn. (16). If Wlz(k ) and F(k) are the amplitudes of w12 and the right-hand side of eqn. (16) at wavenumber k, respectively, eqn. (16) can be written
kZy2w12(k)+ H * w , z ( k ) = r ( k )
The magnitude of w12 at wavenumber k is thus
F(k) w , 2 ( k ) =
kZy2 + H *
At long wavelengths, an increase in k will not change w12 significantly, whereas at shorter scales it will be reduced more noticeably: for k - 1/100 km, doubling g ' results in w l z ( k ) falling by 16%. The w12 forcing of the lower layer is less if y is increased, and the effect is greatest on smaller scales (where altimetry is most effective). Thus in this example, where the stratification of the assimilating model is overestimated, the transfer of information into the subsurface layer is expected to be slower than for a perfect model, and the magnitude of the small-scale features underestimated.
Fig. 8. The lower layer streamfunction (CI =1500 m 2 S - 1 ) o n day 108 of the two-layer integration in Fig. 7.
291
11t
~ 4 E
v
~ 2 r r
-6
E O O o
-3 LU
r r
0 20 40 60 80 100 120 TIME (days)
Fig. 9. Ocean topography (in cm) and ~2 (in m 2 s - 1 ) obtained when perfect data on an optimum orbit repeat cycle are assimilated into a two-layer model in which the non-dimen-
1
sional parameter R has been increased to (2~Rt~th) (initialized as in Fig. 2).
Figure 7 indeed shows that after about 70 days the degraded model becomes less efficient than the perfect model at transferring information vertically; by this time the major features in the lower layer have been spun up, but errors in their magnitude are now noticeable. The lower layer streamfunction after 108 days of assimilation, shown in Fig. 8, is similar to the truth (Fig. 5a) except that the eddies are less intense; on scales of 100 km or so, magnitudes are reduced by about 20%. This is consistent with our mechanism for vertical interpolation discussed earlier.
In our second example, the assimilating model is degraded by increasing R by a factor of 2 '2. Thus the magnitude of the advection terms and the width of the inertial boundary current (which scales as R 1/2) is increased. One might expect that an advective velocity that is in error is likely to compromise severely the ability of the model to extrapolate information forward in time.
Figure 9 shows the r.m.s.e, between the truth and degraded model assimilation streamfunctions. The upper layer r.m.s.e, is consistently higher than when a perfect model is used (compare with Fig. 2a, curve C), and indeed after 80 days, altimetry is no longer able to constrain even the surface flow. The ampli tude of the short period oscillations in the r.m.s.e, curve are a measure of how far the model deviates from truth in active areas between passes, and this increases as time goes on, indicating that the model is incapable of projecting data between repeat times, and is no longer provid- ing useful first guess information.
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293
The lower layer r.m.s.e, shown in Fig. 9 indicates that following an initial rapid transfer of information downward from the surface, accuracy is also lost in the deep flow. Figure 10 shows ~2 at 108 days and confirms that the subsurface flow cannot be successfully inferred by assimilating data at the surface of this degraded model. Once again, eqn. (16) provides the clue. The dominant term on the right-hand side (on eddy scales) is RJ (+1, V2~bl), and so the lower layer will be forced by a Wa2 that is in error even if +1 is perfectly specified.
These results emphasize the importance of the accurate representation of non-linear processes when assimilating into wind-driven models, especially if they are to be used to infer the deep flow. The effects of both model degradations discussed here are consistent with the mechanism for vertical interpolation formulated in the previous section, suggesting that the impact of other model degradations could be assessed by recourse to this theory.
4. 5. Assimilation into multi-layer models
Having enquired into the behaviour of a two-layer model assimilating altimeter data at the surface, we now go on to consider whether a model of higher vertical resolution may be similarly constrained. The three-layer simulation of idealized ocean gyres studied by Marshall et al. (1988) is taken as our reference ocean. The model parameters, presented in the Appendix, were chosen to ensure that the internal jet has a realistic penetration scale. The upper layer (from which the altimeter data are taken, and into which they are assimilated) is 500 m deep, the middle 1000 m, whilst the deep flow is represented by a third layer 3500 m deep. The circulation is driven by a wind-stress curl of the same magnitude and form as in the two-layer integration. However, the higher vertical resolution of the model and the thinner upper layer results in a circulation which is unstable not only in the region of the jet but also in the interior. Figure 11 shows instantaneous streamfunction plots in each of the three layers (from the truth circulation at day 108).
In section 4.4, we described how, in a two-layer model, the deep flow comprises a superposition of essentially linear Rossby waves excited by the interracial vertical velocity, which is controlled almost entirely by the upper layer flow. Errors in the initial conditions in the deep flow propagate as Rossby waves, damped by the Ekman friction on an e-folding time-scale of 1/c. In the three-layer case, the deep third layer will respond to forcing by the interracial velocity between it and the second layer in much the same way. We must first, however, consider the reduction of errors in the initial conditions of the second layer. The surface flow will drive the middle layer through vortex stretching but now the interfacial velocity w12 will depend on
294
(a)
....... ( 5 . . . . . . . . ' ...........
~ ~ ¢ .... .... " :K,. ! ......... i ....
(b) (c) Fig. 11. (a) Upper (b) middle and (c) bottom layer streamfunctions from the truth three-layer model on day 108; CI =4 cm (of ocean topography), 4200 m 2 s -1 and 2100 m 2 s 1 respectively.
the flow in bo th layers, and will not be con t ro l led (as in the two- layer case) by ~b 1. In fact er rors in the second layer m a y well adverse ly affect f low in the surface layer. F u r t h e r m o r e , since the only dissipat ive process in the second layer is a highly scale-selective b i -ha rmon ic fr ict ion, on ly errors on the small (grid) scale will be d a m p e d rapidly.
Ass imi la t ion of da t a in to the three- layer model on a 14-day repeat ( op t imum ) orb i t is not suff icient to cons t ra in the model : t ru th and analysis diverge. Thus , in o rde r to s tudy the mechan i sm of vert ical in t e rpo la t ion in
295
48 t 36 ~ A
q
0 40 8'0 130 160 260 (a) TIME (days)
481 36
~'24 ~
0 4Q 80 120 160 200 (b) TIME (days)
Fig. 12. Error curves (in m 2 s-1) obtained from a three-layer model that has been initialized with the 8-month truth climatology. Curve A is the variance of the truth about climatology, and curve B shows the r.m.s.e, between the model and truth streamfunctions where 'infinite altimetry' has been applied. (a) Second (middle) layer; (b) third (bottom) layer.
this three-layer case, results are presented from an assimilation experiment in which the surface layer is fully constrained by infinite altimetry; the upper layer streamfunction is replaced at all grid points and time steps by its truth value. The model was initialized from the (truth) 8-month time mean climatology. The r.m.s.e, curves (in m 2 s -1) between the streamfunction fields in layers two and three and the truth are shown in Fig. 12, together with the analysis streamfunction fields after 180 days of assimilation in Fig. 13. Note the difference in the vertical scale of Fig. 12(b) and that of Fig. 4 for the two-layer model; the basin-averaged variability in the three-layer model is much greater than in the two-layer case because now the abyssal eddy field almost fills the basin (cf. Fig. l l ( c ) with Fig. 5(a)). From Fig. 12, it can be seen that there has been some reduction in the error in both +2 and +3, although the fall is not nearly so rapid as in the two-layer case. After 180 days, the r.m.s.e, in the second layer has fallen to 50% of its initial value (the e-folding time is approximately 150 days). The deep flow r.m.s.e, falls to 40% of its original value, with an e-folding time of 110 days. In this integration, 1/~ = 74 days, so our model of error reduction as a damped linear Rossby wave does not seem to be appropriate here. Evidently non-lin- earities in the second layer severely inhibit the vertical transfer of informa- tion from the surface to the deep layer. However, it should be stressed that there is still some error reduction: many of the features in both subsurface layers have been inferred correctly, as can be seen by comparing Fig. 13 with Fig. 11.
It can be concluded, then, that if surface pressure data are assimilated into models with more than two layers the ability of the model to spin up
296
7~2
....... i
Fig. 13. Second and th i rd layer s t r e a m f u n c t i o n p lo ts on day 108 wi th ' i n f in i t e a l t imet ry ' for the th ree- layer mode l o f Fig. 11; CI = 4200 m 2 s - l and 2100 m 2 s -1 respect ively.
the deep layers is severely diminished. In the absence of further information a single altimeter cannot constrain a multi-layer model, even when flown in an optimum repeat orbit. It should be emphasized, however, that here no attempt has been made to use statistical techniques (for example vertical structure functions) to extrapolate surface information into deeper layers or dynamical methods (for example, projecting onto vertical modes as pro- posed by Webb and Moore (1986)). Neither has the use of subsurface in situ
observations been considered in conjunction with altimetry. Each of these possibilities deserves further work.
5. D I S C U S S I O N
Our investigation is presented very much in the spirit of the early network studies directed at optimizing and evaluating the meteorological observing network. Rather than focusing on a limited geographical area (as for example in the work reviewed by Robinson (1986), or the study of De Mey and Robinson (1987)) we have considered assimilation into basin-scale models. It is only the advent of radar altimetry that makes this a tenable possibility. It is recognized that in order to obtain an altimetric measure- ment of the ocean topography that is sufficient to determine the mean circulation, a geoid vastly more accurate than is currently available is needed. However, the geoid is being continually refined (indeed, in regions where our knowledge of the ocean circulation is good, the latter can be used
297
in conjunction with altimetry to improve the geoid (see Thompson, 1986)). Here is has been assumed that the geoid is sufficiently well known to allow determination of the surface pressure field from altimetry. We have investi- gated the assimilation into a model of surface pressure field data on space and time-scales obtainable from a single radar altimeter, and assessed the degree to which such information may constrain that model.
The following general conclusions can be drawn. (1) Observations of the surface pressure field on space and time-scales
possible with a single satellite-borne radar altimeter provide strong con- straints on baroclinic, eddy resolving ocean circulation models, both at the surface and in deeper layers.
(2) The skill of the surface is a strong function of the spatial and temporal frequency of the altimetric observations, and is only weakly dependent on the random error in the instrument (for reasonable levels of noise). For synoptic mapping on basin scales, an orbit repeat time of about 14 days is found to be optimum.
(3) The dynamical mechanism by which information at the surface is extrapolated vertically, constraining abyssal flows, can be understood read- ily in two-layer models in which the deep flow is a superposition of damped linear Rossby waves forced by vortex stretching at the interface between the layers. Given a good analysis of the surface pressure field, the forcing of the lower layer can be determined, and so any initial error decays as a damped Rossby wave with an Ekman spin-down time-scale. However, in more realistic models, where many shallow layers are required to represent the strongly depth-dependent currents in the upper levels of the wind-driven gyre, they are less able to spin-up subsurface flows from information provided at the surface.
Finally, it must be stressed that altimetry should be seen as only one, albeit central, component of planned observing networks. Future studies should assess the impact of altimetry in conjunction with both complemen- tary in situ observations (for example, velocity observations at thermocline depth from neutrally buoyant floats) and other remotely sensed information, particularly sea-surface temperature and ocean colour sensors giving the positions of major fronts and eddies.
ACKNOWLEDGEMENT
We would like to thank the staff of the European Centre for Medium Range Weather Forecasts for making available computing facilities and advice, without which this study would not have been possible.
298
APPENDIX
The potential vorticity equation (non-dimensionalized)
1 3qn - - - - +J(~Pn, qn) = Gn (A1) R 3t is integrated forward numerically (Brugge et al., 1987) for each layer (n = 1, 2 or 3) on a finite-difference grid in a square domain, where qn is the quasi-geostrophic potential vorticity, ~Pn is the streamfunction, G n is the potential vorticity forcing function, J(a, b) is the Jacobian of a and b, 3a/Ox Ob/3y - 3b/Ox 3a/3y, and x (0 < x < 1) is east, y ( - 0 . 5 < y < 0.5) is nor th and t is time.
The potential vorticity is related to the streamfunction through
q = Rx7 2~ + y _ Sxi, (A2)
where
/ q = q2 ; 4 = ~2 qa ~kl
and S is the stretching matrix given by
1 - 1 0
31A021 31Ao21 RL 2 - 1 1 1 - 1
- - + - - (A3) S = ~ 32 A021 32 Ao.21 32 A032 32 A032 L r
- 1 1 0
33Ao32 33 A0"32
In Eqns. (A1)-(A3), length has been non-dimensionalized with respect to L, where L is the nor th - sou th extent of the basin; depth with respect to H the depth of the ocean; time with respect to (ilL) -1, where fl is the planetary vorticity gradient; ~p with respect to UsL, where U s is a character- istic velocity chosen to be the Sverdrup velocity U s = foWEK/flH and WEK is an Ekman pumping velocity; q with respect to ilL. fo is the reference Coriolis parameter.
The coefficient R = Us~ilL 2 is a Rossby number for the vorticity equa- tion; Lr=(gH/lOOOfo2) 1/2 is a pseudo-Rossby radius, A%+1, . = 103Apn+l,n/po expresses the density jumps between the layers in o units where
Apn+l,n = Pn + l - - On
and 3 n = H n / H are the fractional layer depths.
299
T h e G s in eqn . (A1) a r e g i v e n b y
1 = Wo - %1
G 2 = -- t-'V 6~ 2
G 3 = - VX7 6~3 -- C~ 2~3
w h e r e W 0 is t he ( n o n - d i m e n s i o n a l i z e d ) v e r t i c a l v e l o c i t y a t t h e b a s e o f t h e
E k m a n l a y e r in u n i t s o f WzK a n d I4/0 is c h o s e n to b e
W 0 = s in 2 t ry
d r i v i n g t h e m o d e l in a d o u b l e - g y r e c o n f i g u r a t i o n . T h e c o e f f i c i e n t s v a n d c
a r e l a t e r a l a n d b o t t o m d i f f u s i o n c o n s t a n t s r e s p e c t i v e l y .
P a r a m e t e r s
G e n e r a l :
T w o - l a y e r :
L = 3 × 10 6 m, 13 = 10 -11 m -1 s -1
H 1 = 1000 m
H 2 = 4000 m
Ao12 = 2 f0 = 7 .0 × 10 - 5 s -1
= 55 d a y s -1
U s = 1.4 × 10 -3 m s -1
R = 1.6 × 1 0 - 5
R o s s b y r a d i u s = 48 k m
T h r e e - l a y e r : H 1 = 500 m
H 2 = 1000 m
H 3 = 3500 m
AO12 = 2
Aa23 = 1 f0 = 7.0 × 10 - 5 s - a e = 74 d a y s - 1
U s = 1.4 × 10 -3 m s -1
R = 1.6 × 10 - 5
R o s s b y r a d i u s = 48 a n d 18 k m
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